Volume 2011, Article ID 368137,15pages doi:10.1155/2011/368137
Research Article
Convergence of Iterative Sequences for Fixed Point
and Variational Inclusion Problems
Li Yu and Ma Liang
School of Management, University of Shanghai for Science and Technology, Shanghai 200093, China
Correspondence should be addressed to Li Yu,[email protected]
Received 14 November 2010; Accepted 8 February 2011 Academic Editor: Yeol J. Cho
Copyrightq2011 L. Yu and M. Liang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
An iterative process is considered for finding a common element in the fixed point set of a strict pseudocontraction and in the zero set of a nonlinear mapping which is the sum of a maximal monotone operator and an inverse strongly monotone mapping. Strong convergence theorems of common elements are established in real Hilbert spaces.
1. Introduction and Preliminaries
Throughout this paper, we always assume that H is a real Hilbert space with the inner product·,·and the norm · .
LetCbe a nonempty closed convex subset ofHandS:C → Ca nonlinear mapping. In this paper, we useFSto denote the fixed point set ofS. Recall that the mappingSis said to be nonexpansive if
Sx−Sy ≤ x−y, ∀x, y∈C. 1.1
Sis said to beκ-strictly pseudocontractive if there exists a constantκ∈0,1such that
Sx−Sy2≤x−y2κx−Sx−y−Sy2, ∀x, y∈C. 1.2
The class of strictly pseudocontractive mappings was introduced by Browder and Petryshyn
LetA:C → Hbe a mapping. Recall thatAis said to be monotone if
Ax−Ay, x−y≥0, ∀x, y∈C. 1.3
Ais said to be inverse strongly monotone if there exists a constantα >0 such that
Ax−Ay, x−y ≥αAx−Ay2, ∀x, y∈C. 1.4
For such a case,Ais also said to beα-inverse strongly monotone.
LetM:H → 2Hbe a set-valued mapping. The setDMdefined byDM {x∈H: Mx /∅}is said to be the domain ofM. The setRMdefined byRM x∈HMxis said to be the range ofM. The setGMdefined byGM {x, y∈H×H:x∈DM, y∈RM} is said to be the graph ofM.
Recall thatMis said to be monotone if
x−y, f−g>0, ∀x, f,y, g∈GM. 1.5
M is said to be maximal monotone if it is not properly contained in any other monotone operator. Equivalently,Mis maximal monotone ifRIrM Hfor allr >0. For a maximal monotone operator M onH and r > 0, we may define the single-valued resolvent Jr
IrM−1:H → DM. It is known thatJr is firmly nonexpansive andM−10 FJr.
Recall that the classical variational inequality problem is to findx∈Csuch that
Ax, y−x≥0, ∀y∈C. 1.6
Denote by VIC, Aof the solution set of1.6. It is known thatx∈Cis a solution to1.6if and only ifxis a fixed point of the mappingPCI−λA, whereλ >0 is a constant andIis the
identity mapping.
Recently, many authors considered the convergence of iterative sequences for the variational inequality1.6and fixed point problems of nonlinear mappings see, for example,
1–32.
In 2005, Iiduka and Takahashi7proved the following theorem.
Theorem IT. Let C be a closed convex subset of a real Hilbert space H. LetA be an α
-inverse-strongly monotone mapping ofCintoH, and letSbe a nonexpansive mapping ofCinto itself such thatFS∩VIC, A/∅. Suppose thatx1x∈Cand{xn}is given by
xn1αnx 1−αnSPCxn−λnAxn, 1.7
for everyn1,2, . . ., where{αn}is a sequence in0,1and{λn}is a sequence ina, b. If{αn}and {λn}are chosen so that{λn} ∈a, bfor somea, bwith0< a < b <2α,
lim
n→ ∞αn0,
∞
n1
αn∞,
∞
n1
|αn1−αn|<∞,
∞
n1
|λn1−λn|<∞, 1.8
In 2007, Y. Yao and J.-C. Yao31further obtained the following theorem.
Theorem YY. Let Cbe a closed convex subset of a real Hilbert space H. Let Abe an α
-inverse-strongly monotone mapping ofCintoH, and letSbe a nonexpansive mapping ofCinto itself such thatFS∩Ω/∅, whereΩdenotes the set of solutions of a variational inequality for theα -inverse-strongly monotone mapping. Suppose thatx1 u∈Cand{xn},{yn}are given by
x1u∈C,
ynPCxn−λnAxn,
xn1αnuβnxnγnSPCI−λnAyn, n≥1,
1.9
where{αn},{βn}, and{γn}are three sequences in0,1and {λn}is a sequence in0,2a. If{αn}, {βn},{γn}, and{λn}are chosen so thatλn∈a, bfor somea, bwith0< a < b <2aand
aαnβnγn1, for alln≥1,
blimn→ ∞αn0, ∞n1αn∞,
c0<lim infn→ ∞βn≤lim supn→ ∞βn<1,
dlimn→ ∞λn1−λn 0,
then{xn}converges strongly toPFS∩Ωu.
In this work, motivated by the above results, we consider the problem of finding a common element in the fixed point set of a strict pseudocontraction and in the zero set of a nonlinear mapping which is the sum of a maximal monotone operator and a inverse strongly monotone mapping. Strong convergence theorems of common elements are established in real Hilbert spaces. The results presented in this paper improve and extend the corresponding results announced by Iiduka and Takahashi7and Y. Yao and J.-C. Yao31.
In order to prove our main results, we also need the following lemmas.
Lemma 1.1see22. LetCbe a nonempty closed convex subset of a Hilbert spaceH, A:C → H
a mapping, andM:H → 2Ha maximal monotone mapping. Then,
FJrI−rA AM−10, ∀r >0. 1.10
Lemma 1.2see1. LetCbe a nonempty closed convex subset of a real Hilbert spaceHandS :
C → Caκ-strict pseudocontraction with a fixed point. DefineS:C → CbySaxax 1−aSx for eachx∈C. Ifa∈κ,1, thenSais nonexpansive withFSa FS.
Lemma 1.3see25. LetCbe a nonempty closed convex subset of a Hilbert spaceHandS:C →
Caκ-strict pseudocontraction. Then,
aSis1κ/1−κ-Lipschitz,
Lemma 1.4see28. Let{xn}and{yn}be bounded sequences in a Hilbert spaceH, and let{βn} be a sequence in0,1with
0<lim inf
n→ ∞ βn≤lim supn→ ∞ βn<1. 1.11
Suppose thatxn1 1−βnynβnxnfor all integersn≥1and
lim sup
n→ ∞
yn1−yn − xn1−xn
≤0. 1.12
Then,limn→ ∞yn−xn0.
Lemma 1.5see29. Assume that{αn}is a sequence of nonnegative real numbers such that
αn1≤
1−γn
αnδn, 1.13
where{γn}is a sequence in0,1and{δn}is a sequence such that
a ∞n1γn∞,
blim supn→ ∞δn/γn≤0or ∞n1|δn|<∞. Then,limn→ ∞αn0.
Lemma 1.6see24. LetHbe a Hilbert space andMa maximal monotone operator onH. Then,
the following holds:
Jrx−Jsx2 ≤ r−x
r Jrx−Jsx, Jrx−x, ∀s, t >0, x∈H, 1.14
whereJr IrM−1andJs IsM−1.
2. Main Results
Theorem 2.1. LetHbe a real Hilbert spaceHandCa nonempty close and convex subset ofH. Let
M : H → 2H andW : H → 2H two maximal monotone operators such thatDM ⊂ Cand DW⊂C, respectively. LetS:C → Cbe aκ-strict pseudocontraction,A:C → Hanα-inverse strongly monotone mapping, andB:C → Haβ-inverse strongly monotone mapping. Assume that F :FS∩AM−10∩BW−10/∅. Let{xn}be a sequence generated in the following manner:
x1∈C,
ynJsnxn−snBxn,
xn1αnuβnxnγn
δnJrn
yn−rnAyn
1−δnSJrn
yn−rnAyn
, ∀n≥1,
2.1
where u ∈ C is a fixed element, Jrn I rnM
−1 and J
sn I snW
−1, {r
Assume that the following restrictions are satisfied:
a0< a≤rn≤b <2α,limn→ ∞rn−rn1 0,
b0< c≤sn≤d <2βn,limn→ ∞sn−sn1 0,
c0≤κ≤δn< e <1,limn→ ∞δn−δn1 0,
dlimn→ ∞αn0, ∞n1αn∞,
e0<lim infn→ ∞βn≤lim infn→ ∞βn<1.
Then, the sequence{xn}converges strongly toqPFu.
Proof. The proof is split into five steps.
Step 1. Show that{xn}is bounded.
Note thatI−rnAandI−snBare nonexpansive for each fixedn≥1. Indeed, we see
from the restrictionathat
I−rnAx−I−rnAy2
x−y2−2rn
x−y, Ax−Ayrn2Ax−Ay2
≤x−y2−rn2α−rnAx−Ay2
≤x−y2, ∀x, y∈C.
2.2
This shows thatI−rnAis nonexpansive for each fixedn≥1, so isI−snB. Put
Snxδnx 1−δnSx, ∀x∈C. 2.3
In view of the restrictionc, we obtain fromLemma 1.2thatSnis a nonexpansive mapping
with FSn FS for each fixed n ≥ 1. Fixing p ∈ F and since Jrn and I − rnA are nonexpansive, we see that
xn1−p ≤αnu−pβnxn−pγnSnJrn
yn−rnAyn
−p
≤αnu−pβnxn−pγnJrn
yn−rnAyn
−p
≤αnu−pβnxn−pγnyn−p ≤αnu−p 1−αnxn−p.
2.4
By mathematical inductions, we see that {xn} is bounded and so is {yn}. This completes
Step 2. Show thatxn1−xn → 0 asn → ∞.
Notice fromLemma 1.6that
yn1−yn ≤ xn1−sn1Bxn1−xn−snBxn
Jsn1xn−snBxn−Jsnxn−snBxn
≤ xn1−xn|sn1−sn|Bxn
|sn1−sn| sn1
Jsn1xn−snBxn−xn−snBxn
≤ xn1−xn2M1|sn1−sn|,
2.5
whereM1is an appropriate constant such that
M1 max
sup
n≥1
{Bxn},sup n≥1
J
sn1xn−snBxn−xn−snBxn sn1
. 2.6
Put
znJrn
yn−rnAyn
, ∀n≥1. 2.7
In a similar way, we can obtain fromLemma 1.6that
zn1−zn ≤
yn1−rn1Ayn1
−yn−rnAyn
Jrn1
yn−rnAyn
−Jrn
yn−rnAyn
≤yn1− −yn|rn1−rn|Ayn
|rn1−rn| rn1
Jrn1
yn−rnAyn
−yn−rnAyn
≤ yn1−yn2M2|rn1−rn|,
2.8
whereM2is an appropriate constant such that
M2 max
sup
n≥1
Ayn
,sup
n≥1
Jrn1
yn−rnAxn
−yn−rnAyn
rn1
. 2.9
Substituting2.5into2.8yields that
whereM3is an appropriate constant such that
M3max{2M1,2M2}. 2.11
It follows from2.10that
Sn1zn1−Snzn ≤ zn1−znzn−Szn|δn−δn1|
≤ xn1−xnM4|sn1−sn||rn1−rn||δn−δn1|,
2.12
whereM4is an appropriate constant such that
M4 max
sup
n≥1
{zn−Szn}, M3
. 2.13
Put
ln
xn1−βnxn
1−βn
, ∀n≥1. 2.14
Note that
ln1−ln
αn1uγn1Sn1zn1
1−βn1 −
αnuγnSnzn
1−βn
αn1
1−βn1 −
αn
1−βn
u γn1 1−βn1
Sn1zn1−Snzn
γ
n1
1−βn1 −
γn
1−βn
Snzn
αn1
1−βn1 −
αn
1−βn
u−Snzn
γn1
1−βn1
Sn1zn1−Snzn.
2.15
It follows from2.12that
ln1−ln ≤
αn1
1−βn1 −
αn
1−βn
u−Snzn γn1
1−βn1
Sn1zn1−Snzn
≤ αn1
1−βn1 −
αn
1−βn
u−Snznxn1−xn
M4|sn1−sn||rn1−rn||δn−δn1|.
2.16
This in turn implies from the restrictionsa–ethat
lim sup
FromLemma 1.4, we obtain that
lim
n→ ∞ln−xn0. 2.18
Notice that
xn1−xn
1−βn
ln−xn. 2.19
It follows that
lim
n→ ∞xn1−xn0. 2.20
This completesStep 2.
Step 3. Show thatxn−Sxn → 0 asn → ∞.
SinceJrnandJsn are nonexpansive, we see that
zn−p2≤
xn−p2−rn2α−rnAyn−Ap2, 2.21
yn−p2≤xn−p2−sn
2β−snBxn−Bp2. 2.22
It follows from2.21that
xn1−p2≤αnu−p2βnxn−p2γnSnzn−p2
≤αnu−p2βnxn−p2γnzn−p2
≤αnu−p2xn−p2−γnrn2α−rnAyn−Ap2.
2.23
This in turn implies that
γnrn2α−rnAyn−Ap2≤αnu−p2xn−xn1xn−pxn1−p
.
2.24
In view of2.20, we see from the restrictionsa,d, andethat
lim
It follows from2.22that
xn1−p2≤
αnu−p2βnxn−p2γnSnzn−p2
≤αnu−p2βnxn−p2γnJrnyn−rnAyn−p
2
≤αnu−p2βnxn−p2γnyn−p2
≤αnu−p2xn−p2−γnsn
2β−snBxn−Bp2.
2.26
This in turn implies that
γnsn
2β−snBxn−Bp2≤αnu−p2xn−xn1xn−pxn1−p
.
2.27
In view of2.20, we see from the restrictionsa,d, andethat
lim
n→ ∞Bxn−Bp0. 2.28
SinceJrnis firmly nonexpansive, we obtain that
zn−p2
Jrnyn−rnAyn−Jrnp−rnAp
2
≤zn−p,
yn−rnAyn
−p−rnAp
1
2
zn−p2yn−rnAyn−p−rnAp2
−zn−p
−yn−rnAyn
−p−rnAp2
≤ 1
2
zn−p2yn−p2−zn−ynrn
Ayn−Ap2
1
2
zn−p2yn−p2−zn−yn2−rn2Ayn−Ap2
−2rn
zn−yn, Ayn−Ap
≤ 1
2
zn−p2yn−p2−zn−yn22rnzn−ynAyn−Ap
≤ 1
2
zn−p2xn−p2−zn−yn22rnzn−ynAyn−Ap
.
2.29
This in turn implies that
zn−p2≤
In a similar way, we can obtain that
yn−p2≤
xn−p2−yn−xn22snyn−xnBxn−Bp. 2.31
In view of2.30, we see that
xn1−p2≤
αnu−p2βnxn−p2γnSnzn−p2
≤αnu−p2βnxn−p2γnzn−p2
≤αnu−p2xn−p2−γnzn−yn22rnzn−ynAyn−Ap.
2.32
It follows that
γnzn−yn2≤αnu−p2xn−xn1
xn−pxn1−p
2rnzn−ynAyn−Ap.
2.33
In view of2.25, we obtain from the restrictionsdandethat
lim
n→ ∞zn−yn0. 2.34
Notice from2.31, we see that
xn1−p2≤αnu−p2βnxn−p2γnSnzn−p2
≤αnu−p2βnxn−p2γnzn−p2
≤αnu−p2βnxn−p2γnyn−p2
≤αnu−p2xn−p2−γnyn−xn22snyn−xnBxn−Bp.
2.35
It follows that
γnyn−xn2≤αnu−p2xn−xn1xn−pxn1−p
2snyn−xnBxn−Bp.
2.36
In view of2.28, we obtain from the restrictionsdandethat
lim
Combining2.34with2.37yields that
lim
n→ ∞zn−xn0. 2.38
Note that
xn1−xnαnu−xn γnSnzn−xn. 2.39
In view of2.20, we see from the restrictiondthat
lim
n→ ∞Snzn−xn0. 2.40
Note that
Szn−xn
Snzn−xn
1−δn
δnxn−zn
1−δn
. 2.41
From2.38and2.40, we get from the restrictioncthat
lim
n→ ∞Szn−xn0. 2.42
Notice that
Sxn−xn ≤ Sxn−SznSzn−xn. 2.43
In view of2.38and2.42, we see fromLemma 1.3that
lim
n→ ∞Sxn−xn0. 2.44
This completesStep 3.
Step 4. Show that lim supn→ ∞u−q, xn−q ≤0, whereqPFu.
To show it, we may choose a subsequence{xni}of{xn}such that
lim sup
n→ ∞
u−q, xn−q
lim sup
i→ ∞
u−q, xni−q
. 2.45
Since{xni}is bounded, we can choose a subsequence{xnij}of{xni}converging weakly to
x. We may, without loss of generality, assume that xni x, where denotes the weak convergence. Next, we prove thatx∈ F. In view of2.44, we can conclude fromLemma 1.3
thatx∈FSeasily. Notice that
Letμ∈Mν. SinceMis monotone, we have
y
n−zn
rn −
Ayn−μ, zn−ν
≥0. 2.47
In view of the restrictiona, we see from2.34that
−Ax−μ, x−ν≥0. 2.48
This implies that−Ax ∈Mx, that is,x ∈AM−10. In similar way, we can obtain that x∈BW−10. This proves thatx∈ F. It follows from2.45that
lim sup
n→ ∞
u−q, xn−q
≤0. 2.49
This completesStep 4.
Step 5. Show thatxn → qasn → ∞.
Notice that
xn1−q2αnu−q, xn1−qβn
xn−q, xn1−q
γn
SnJrn
yn−rnAyn
−q, xn1−q
≤αnu−q, xn1−q
βn
2
xn−q2xn1−q2
γn
2
SnJrn
yn−rnAyn
−q2xn1−q2
≤αn
u−q, xn1−q
βn
2
xn−q2xn1−q2
γn
2
yn−q2xn1−q2
≤αn
u−q, xn1−q
1−αn
2
xn−q2xn1−q2
.
2.50
This in turn implies that
xn1−q2≤
1−αnxn−q22αn
u−q, xn1−q
. 2.51
In view of2.49, we conclude fromLemma 1.5that
lim
n→ ∞xn−q0. 2.52
If S is a nonexpansive mapping and δn 0, then Theorem 2.1 is reduced to the
following.
Corollary 2.2. LetHbe a real Hilbert spaceHandCa nonempty close and convex subset ofH. Let
M :H → 2HandW :H → 2Hbe two maximal monotone operators such thatDM⊂ Cand DW ⊂ C, respectively. LetS : C → Cbe a nonexpansive mapping,A : C → Hanα-inverse strongly monotone mapping andB:C → Haβ-inverse strongly monotone mapping. Assume that F :FS∩AM−10∩BW−10/∅. Let{xn}be a sequence generated in the following manner:
x1∈C,
ynJsnxn−snBxn,
xn1αnuβnxnγnSJrn
yn−rnAyn
, ∀n≥1,
2.53
whereu ∈ Cis a fixed element,Jrn I rnM
−1 and J
sn I snW
−1,{r
n}is a sequence in
0,2α,{sn}is a sequence in0,2βand{αn},{βn}and{γn}are sequences in0,1. Assume that the following restrictions are satisfied:
a0< a≤rn≤b <2α, limn→ ∞rn−rn1 0,
b0< c≤sn≤d <2βn, limn→ ∞sn−sn1 0,
climn→ ∞αn0, ∞n1αn∞,
d0<lim infn→ ∞βn≤lim infn→ ∞βn<1.
Then, the sequence{xn}converges strongly toqPFu.
Next, we consider the problem of finding common fixed points of three strict pseudocontractions.
Theorem 2.3. LetCbe a nonempty closed convex subset of a real Hilbert spaceHandPCthe metric
projection fromHontoC. LetS:C → Cbe aκ-strict pseudocontraction,TA:C → Hanα-strict pseudocontraction, andB:C → Haβ-strict pseudocontraction. Assume thatF:FS∩FTA∩ FTB/∅. Let{xn}be a sequence generated in the following manner:
x1∈C,
zn 1−snxnsnTBxn,
yn 1−rnznrnTAzn
xn1 αnuβnxnγn
δnyn 1−δnSyn
, ∀n≥1,
2.54
whereu∈Cis a fixed element,{rn}is a sequence in0,1−α,{sn}is a sequence in0,1−β, and {αn},{βn},{γn}, and{δn}are sequences in0,1. Assume that the following restrictions are satisfied
a0< a≤rn≤b <1−α, limn→ ∞rn−rn1 0,
b0< c≤sn≤d <1−βn, limn→ ∞sn−sn1 0,
dlimn→ ∞αn0, ∞n1αn∞,
e0<lim infn→ ∞βn≤lim infn→ ∞βn<1.
Then, the sequence{xn}converges strongly toqPFu.
Proof. PuttingA I−TA, we see thatAis1−α/2-inverse strongly monotone. We also
haveFTA VIC, AandPCxn−rnAxn 1−rnxnrnTxn. PuttingBI−TB, we see that Bis1−β/2-inverse strongly monotone. We also haveFTB VIC, BandPCxn−snBxn
1−snxnsnRun. In view ofTheorem 2.1, we can obtain the desired results immediately.
Acknowledgments
The authors are extremely grateful to the referees for useful suggestions that improved the contents of the paper. This work was supported by the National Natural Science Foundation of China under Grant no. 70871081 and Important Science and Technology Research Project of Henan province, China102102210022.
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